Problem 790

Erdős Problem #790: declared status 'open'. Formalized: no. Let $l(n)$ be maximal such that if $A\subset\mathbb{Z}$ with $\lvert A\rvert=n$ then there exists a sum-free $B\subseteq A$ with $\lvert B\rvert \geq l(n)$ - that is, $B$ is such that there are no solutions to$$a_1=a_2+\cdots+a_r$$with $a_i\in B$ all distinct. Estimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\to \infty$? Is it true that $l(n)< n^{1-c}$ for some $c>0$? Prize: no. Tags: additive combinatorics.